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This is a symmetric companion to the author's non-symmetric polar-degree preprint (arXiv:7680505); the method parallels that work, but the proof here is self-contained and redoes the load-bearing local incidence analysis in the symmetric setting. The general theorem: if X = V(f) in P^{N-1} is a smooth degree-d hypersurface, N >= 3, and f = det(A_0 + sum "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.11090","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2026-06-09T16:49:37Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"8e21ce2d1f4376d9f903fee0e8163af28404abf7f9bfa854b605368f51c7c805","abstract_canon_sha256":"9c849b2709cf0532f9ad44269b856db48af5c35a93db7dee9d7e732ab8599122"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:11:09.617614Z","signature_b64":"T57Qx4t9SB8wiejzAVn2FHxdnFbmGbXZ9iU48E/xwRcwevVwyUpoeLSWyeMDxoiLc2mpUMECFrED7rTi3IXzDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"abfec8f61b7214695b6ca8042d52b19da3bbec321299d900ee6e7f964f399d4e","last_reissued_at":"2026-06-10T01:11:09.616777Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:11:09.616777Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A symmetric determinantal lower bound for diagonal power sums via polar degree","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"cs.CC","authors_text":"Karthik Sheshadri","submitted_at":"2026-06-09T16:49:37Z","abstract_excerpt":"The symmetric determinantal complexity sdc(f) of a polynomial f is the least m such that f = det(M) for an m x m symmetric matrix M of affine-linear forms. 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